Correct, that 'someone' doesn't satisfy the definition of having faith. They don't have it because they've rejected it, not because God didn't convict them. God calls them to himself through the conviction of faith but instead of allowing that conviction of faith to lead them to believing/trusting in Christ they reject it instead.
To 'have' faith means to accept the convicting testimony of the Holy Spirit. That's what the believer does. He accepts the faith God is giving to believe and be saved. God presents the unbeliever with the faith to believe, too, but they reject it, not accept it. As a result they do not have faith. How can they? They've rejected it.